The substring reversal graph Rn is the graph whose vertices are the permutations Sn, and where two permutations are adjacent if one is obtained from a substring reversal of the other. We determine the spectral gap of Rn, and show that its spectrum contains many integer values. Further we consider a family of graphs that generalize the prefix reversal (or pancake flipping) graph, and show that e...

This is my spin of the some stuff from from [1]. CONTENTS 1. Double complexes 1 2. The generalized Mayer-Vietoris principle 6 3. Presheaves and sheaves 17 4. The Čech cohomology of presheaves 22 5. Spectral sequences 25 6. The Leray-Serre spectral sequence 32 References 34 1. DOUBLE COMPLEXES Suppose that R is a commutative ring with 1. A double complex of R-modules is a bigraded R-module E•,• ...

Let M be a connected, noncompact, complete Riemannian manifold, consider the operator L = ∆+∇V for some V ∈ C(M) with exp[V ] integrable w.r.t. the Riemannian volume element. This paper studies the existence of the spectral gap of L. As a consequence of the main result, let ρ be the distance function from a point o, then the spectral gap exists provided limρ→∞ supLρ < 0 while the spectral gap d...

Article history: Received 19 September 2007 Received in revised form 10 April 2008 Accepted 24 June 2008

Spectral clustering has attracted much research interest in recent years since it can yield impressively good clustering results. Traditional spectral clustering algorithms first solve an eigenvalue decomposition problem to get the low-dimensional embedding of the data points, and then apply some heuristic methods such as k-means to get the desired clusters. However, eigenvalue decomposition is...

We present a new clustering algorithm that is based on searching for natural gaps in the components of the lowest energy eigenvectors of the Laplacian of a graph. In comparing the performance of the proposed method with a set of other popular methods (KMEANS, spectral-KMEANS, and an agglomerative method) in the context of the Lancichinetti-Fortunato-Radicchi (LFR) Benchmark for undirected weigh...

of the Dissertation Spectral Clustering for Complex Settings Many real-world datasets can be modeled as graphs, where each node corresponds to a data instance and an edge represents the relation/similarity between two nodes. To partition the nodes into different clusters, spectral clustering is used to find the normalized minimum cut of the graph (in the relaxed sense). As one of the most popul...

As a novel clustering algorithm, spectral clustering is applied in machine learning extensively. Spectral clustering is built upon spectral graph theory, and has the ability to process the clustering of non-convex sample spaces. Most of the existing spectral clustering algorithms are based on k-means algorithm, and k-means algorithm uses the iterative optimization method to find the optimal sol...